In the proof for Euler's formula, we expand $e^{ix}$ as a Taylor series, rearrange the terms, factor out $i$, and thus obtain the Taylor series for $\sin (x)$ and $\cos(x)$. However, this rearrangement can only be done if the Taylor series for $e^{ix}$ is absolutely convergent, by the Riemann series theorem.
I know how to prove that a series of real terms is absolutely convergent. However, how do you do the same for a series of complex terms, like the one obtained in the Taylor series expansion of $e^{ix}$?